An important problem for HCI researchers is to estimate the parameter values of a cognitive model from behavioral data. This is a difficult problem, because of the substantial complexity and variety in human behavioral strategies. We report an investigation into a new approach using approximate Bayesian computation (ABC) to condition model parameters to data and prior knowledge. As the case study we examine menu interaction, where we have click time data only to infer a cognitive model that implements a search behaviour with parameters such as fixation duration and recall probability. Our results demonstrate that ABC (i) improves estimates of model parameter values, (ii) enables meaningful comparisons between model variants, and (iii) supports fitting models to individual users. ABC provides ample opportunities for theoretical HCI research by allowing principled inference of model parameter values and their uncertainty.